On the structure of Goldie Modules

0.2 MB·

by  Jaime Castro Pérez

ON THE STRUCTURE OF GOLDIE
6
1 MODULES
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2

n
a Jaime Castro P´erez∗
J
Escuela de Ingenier´ıa y Ciencias, Instituto Tecnol´ologico y de
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1
Estudios Superiores de Monterrey

] Calle del Puente 222, Tlalpan, 14380, M´exico D.F., M´exico.
A
R Mauricio Medina B´arcenas† Jos´e R´ıos Montes‡
h. Angel Zald´ıvar§
t
a Instituto de Matem´aticas, Universidad Nacional
m
Auto´noma de M´exico
[

Area de la Investigacio´n Cient´ıfica, Circuito Exterior, C.U.,
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v
04510, M´exico D.F., M´exico.
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4
4 January 15, 2016
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.
1
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Abstract
1
:
v Given a semiprime Goldie module M projective in σ[M] we study
i
X decompositions on its M-injective hull M in terms of the minimal
r prime in M submodules. With this, we characterize the semiprime
a
Goldie modules in Z-Mod and make a deccomposition of the endomor-
phism ring of M. Also, we investigate the relations among semiprime
Goldie modules, QI-modules and co-semisimple modules extending
results on leftcQI-rings and V-rings.
∗[email protected],correspondingauthor
†[email protected]
‡[email protected]
§[email protected]
1
1 Introduction and preliminars
In this paper we study in a deeper way the Goldie modules structure. Goldie
Modules where introduce in [3] as a generalization of left Goldie ring. A
Goldie module is such that it satisfies ACC on left annihilators and has fi-
nite uniform dimension. In [2] are studied, in general modules which satisfy
ACC on left annihilators; in this article the authors show that a module M
projective in σ[M] and semiprime which satisfies ACC on left annihilators
has finitely many minimal prime submodules [2, Theorem 2.2]. Using this
fact, if M is a semiprime Goldie module projective in σ[M] we study in
this manuscript the relation between the M-injective hull of M and the M-
injective hulls of the factor modules given by the minimal prime submodules
of M. We prove in Corollary 2.7.1 that the M-injective hull is isomorphic
to the direct sum of the M-injective hulls of the factor modules given by the
minimal prime submodules of M. Moreover the summands of this decompo-
sition are homological independent.
In [3, Proposition 2.25] is proved that if M is a semiprime Goldie module
projective in σ[M] with P ,...,P its minimal prime submodules the each
1 n
M/P is a prime Goldie module. With that, we prove in Theorem 2.19 that
i
the endomorphism ring of the M-injective hull of M is a direct product of
simple artinian rings and each simple artinian ring is a right quotient ring.
Weinvestigatewhenaquasi-injectivemoduleinσ[M]isinjectiveprovided
M is a semiprime Goldie module, this in order to prove when a semiprime
Goldie module is a QI-module and generalize some results given by Faith
in [9].
Talking about QI-modules arise co-semisimple modules. We give some
results on co-semisimple Goldie modules, we show that every indecompos-
able injective summand of the M-injective hull of a co-semisimple Goldie
module is FI-simple Proposition 4.11, also we see that every co-semisimple
prime Goldie module is FI-simple Proposition 4.6. This gives as consequence
that if M is a co-semisimple module then M has finitely many prime sub-
modules and every factor M/N with N a fully invariant submodule of M is
a semiprime Goldie module.
We organized the content of this paper as follows: Fist section is this
introduction and the necessary preliminars for the develop of this theory.
Secondsectionconcernstogivesomeresultsonthestructureofsemiprime
Goldiemodules, wedothisshowing somedecompositionsintheirM-injective
hulls. Also, we show examples of semiprime Goldie modules and characterize
2
all of them in Z-Mod.
Insectionthreewestudyquasi-injectivemodulesinσ[M]andQI-modules.
We give conditions on a semiprime Goldie module in order to every quasi-
injective module to be injective in σ[M]. Also at the end of this section we
give some observations on hereditary semirime Goldie modules.
The last section describe co-semisimple Goldie modules, here is shown
that a co-semisimple Goldie module is isomorphic to a direct product of
prime Goldie FI-simple modules.
ThroughthispaperRwilldenoteanassociativeringwithunitaryelement
and all R-modules are left unital modules. R-Mod denotes the category of
left R-modules. Fora submodule N of a moduleM we write N ≤ M if N is a
propersubmodule we just writeN < M andif itisessential N ≤ M. Afully
e
invariant submodule N of M, denoted N ≤ M is a such that f(N) ≤ N
FI
for all f ∈ End (M). When a module has no nonzero fully invariant proper
R
submodules it is called FI-simple.
Remember that if M is an R-module, σ[M] denotes the full subcategory
R
of R-Mod consisting of those modules M-subgenerated (see [16]). To give
a better develop of the content, for X an R-module and K ∈ σ[X] we will
denote by E[X](K) the X-injective hull of K. When X = K we just write
E[X](X) = X.
A module X ∈ σ[M] is called M-singular if there exists a exact sequence
b
0 → K → L → X → 0
such that K ≤ L. Every module N in σ[M] has a largest M-singular
e
submodule Z(N), if Z(N) = 0 then we say that N is non M-singular.
Given two submodules N,K ≤ M, in [10] is defined the product of N
and K in M as
N K = {f(N)|f ∈ Hom (M,K)}
M R
X
In [6, Proposition 1.3] can be found some proprieties of this product. In
general, this product is not associative but if M is projective in σ[M] it
is [1, Proposition 5.6]. Moreover
Lemma 1.1. Let M be projective in σ[M]. Let K ≤ M and {X } a family
i I
of modules in σ[M]. Then
K X = (K X )
M i M i
!
I
X X
3
Proof. By [6, Proposition 1.3] we have that (K X ) ≤ K ( X ).
M i M I i
Now, letf : M → X . Wehave acanonical epimorphism g : X →
I i P P I i
X . Since M is projective in σ[M] there exist {f : M → X } such that
I i P i i I L
g ◦ f = f. Hence
P I i
L
f(K) = (g ◦ f )(K) = f (K) ≤ (K X )
i i M i
I
M X X
Thus K ( X ) ≤ (K X ).
M I i M i
WiththPisproduct,Pin[13]and[14]theauthorsdefineprimeandsemiprime
submodule respectively in the following way
Definition 1.2. Let M be an R-module and P ≤ M. It is said P is a
FI
prime submodule in M if whenever N K ≤ P with N and K fully invariant
M
submodules of M then N ≤ P or K ≤ P. If 0 is prime in M we say M is a
prime module.
Definition 1.3. Let M be an R-module and N ≤ M. It is said N is a
FI
semiprime submodule in M if whenever K K ≤ P with K a fully invariant
M
submodules of M then K ≤ P. If 0 is semiprime in M we say M is a
semiprime module.
Given K,X ∈ σ[M], the annihilator of K in X is defined as
Ann (K) = {Ker(f)|f ∈ Hom (X,K)
X R
\
This submodule has the propriety that is the greatest submodule of X such
that Ann (K) K = 0. If we consider K ≤ M with M projective in σ[M]
M X
then can be defined the right annihilator of K as
Annr (K) = {L ≤ M|K L = 0}
M M
X
In [2, Remark 1.15] is proved that Annr (K) is a fully invariant submodule
M
of M and is the greatest submodule of M such that K Annr (K) = 0. One
M M
advantageofworkingwithsemiprimemodulesisthatAnn (K) = Annr (K)
M M
for all K ≤ M ( [2, Proposition 1.16]).
Following [4]
4
Definition 1.4. Let M ∈ R−Mod. We call a left annihilator in M a
submodule
A = {Ker(f)|f ∈ X}
X
for some X ⊆ End (M). \
R
As we mentioned before, in [2, Theorem 2.2] was proved that a semiprime
moduleM projectiveinσ[M]andwhichsatisfiesACConleftannihilatorshas
finitely many minimal prime submodules; call these P ,...,P . The minimal
1 n
primes gives a decomposition of M in the following way
M = E[M](Nc1)⊕···⊕E[M](Nn)
where Ni = AnnM(Pi)c.
In the next section we present a decomposition of M equivalent to the
last one in the sense of Krull-Remak-Schmidt, using the M-injective hulls of
M/P . c
i
2 A Decomposition of a Goldie Module
Lemma 2.1. Let M be projective in σ[M] and semiprime. Suppose M sat-
isfies ACC on left annihilators and P ,...,P are the minimal prime in M
1 n
submodules. If N = Ann (P ) for each 1 ≤ i ≤ n then N is a pseudocom-
i M i i
plement of N .
j
j6=i
L
Proof. It is enough to prove that N is a p.c. of n N . Let L ≤ M such
1 i=2 i
that L ∩ ( n N ) = 0 and N ≤ L. Since n N is a fully invariant
submodule thie=n2 ( i n N ) L ≤1( n N )∩L =Li0=.2Sini ce N L ≤ N then
n (N LL) = 0. Hi=e2ncei NM L = 0i=fo2r aill 2 ≤Li ≤ n. So, biyM[2, Theiorem
i=2 iM L iM L
2.2] and [3, Proposition 1.22], L ≤ Ann (N ) = Ann (Ann (P )) = P for
M i M M i i
Lall 2 ≤ i ≤ n. Then L ≤ n P = Ann (P ) = N . Thus L = N .
i=2 i M 1 1 1
Corollary 2.2. Let M bTe projective in σ[M] and semiprime. Suppose M
satisfies ACC on left annihilators and P ,...,P are the minimal prime in M
1 n
submodules. If N = Ann (P ) for each 1 ≤ i ≤ n then M = E[M](N ) ⊕
i M i 1
···E[M](N ).
n
c
Proof. By Lemma 2.2 n N ≤ M. Thus we have done.
i=1 i e
L
5
Lemma 2.3. Let M be projective in σ[M] and semiprime. Suppose that
M satisfies ACC on left annihilators and P ,...,P are the minimal prime
1 n
in M submodules. If N = Ann (P ) for each 1 ≤ i ≤ n then P is a
i M i i
pseudocomplement of N which contains N for all 1 ≤ i ≤ n. Moreover
i j
j6=i
N ≤ P for all 1 ≤ i ≤ n. L
j e i
j6=i
L
Proof. Fix 1 ≤ i ≤ n. Let L ≤ M such that P ≤ L and L ∩ N = 0.
i i
Since N is a fully invariant submodule of M then N L ≤ N ∩L = 0. So
i iM i
L ≤ Ann (N ) = Ann (Ann (P )) = P . Thus L = P .
M i M M i i i
Now by [2, Lemma 2.25] we have that N = Ann (P ) = ∩P . Hence
i M i j
i6=j
N ⊆ P for all j 6= i. Then N ≤ P and by Lemma 2.2, N ≤ P .
j i j i j e i
j6=i j6=i
L L
Lemma 2.4. Let M be projective in σ[M] and semiprime. Suppose that M
satisfies ACC on left annihilators and P ,P ,...,P are the minimal prime in
1 2 n
M submodules. If N = Ann (P ) for 1 ≤ i ≤ n, then P +N ⊆ M for all
i M i i i e
1 ≤ i ≤ n.
Proof. By Lemmas 2.2 and 2.3, N and P are psudocomplements one each
i i
other for all 1 ≤ i ≤ n, thus N +P = N ⊕P ≤ M for all 1 ≤ i ≤ n.
i i i i e
Lemma 2.5. Let M be projective in σ[M] and semiprime. Suppose that M
satisfies ACC on left annihilators and P ,P ,...,P are the minimal prime in
1 2 n
M submodules. If N = Ann (P ) for 1 ≤ i ≤ n, then
i M i
P = M ∩ E[M](N )
i j
j6=i
M
for all 1 ≤ i ≤ n.
Proof. ByCorollary2.2,M = E[M](N )⊕...⊕E[M](N ). ByLemma2.4,M =
1 n
E[M](P )⊕E[M](N ) for each 1 ≤ i ≤ n, moreover E[M](P ) = E[M](N )
i i i j6=i j
by Lemma 2.3. c c
L
We have that N ∩(M∩E[M](P )) = N ∩E[M](P ) = 0, since P is p.c. of
i i i i i
N then P = M∩E[M](P ). Thus P = M∩ E[M](N ) for all 1 ≤ i ≤ n.
i i i i j
j6=i
L
Proposition 2.6. Let M be projective in σ[M] and semiprime. Suppose that
M satisfies ACC on left annihilators and P ,P ,...,P are the minimal prime
1 2 n
in M submodules. If N = Ann (P ) for 1 ≤ i ≤ n, then the morphism
i M i
Ψ : M → M/P ⊕M/P ⊕...⊕M/P
1 2 n
6
givenbyΨ(m) = (m+P ,m+P ....,m+P ) is monomorphismand ImΨ ⊆
1 2, n e
n
M/P .
i
i=1
L
n
Proof. By [3, Corollary 1.15] we have that, ∩P = 0. Thus Ψ is monomor-
i
i=1
n
phism. Now let 0 6= (m +P ,m +P ....,m +P ) ∈ M/P . By 2.4
1 1 2 2, n n i
i=1
n L
P + ∩P ⊆ Mn. So there exists r ∈ R such that
i j e
i=1(cid:18) i6=j (cid:19)
L
n
0 6= r(m ,m ....,m ) ∈ P + ∩P
1 2, n i j
i6=j
i=1 (cid:18) (cid:19)
M
Hence rm ∈ P + ∩P for 1 ≤ i ≤ n. Thus there exists x ∈ P and
i i j i i
i6=j
y ∈ ∩P such that rm = x + y for every 1 ≤ i ≤ n. We claim that
i j i i i
i6=j
r(m +P ,m +P ....,m +P ) ∈ ImΨ. In fact let m = y +y +...+y ,
1 1 2 2, n n 1 2 n
then
Ψ(m) = (y +y +...+y +P ,....,y +y +...+y +P )
1 2 n 1 1 2 n n
= (y +P ,....,y +P ) = (rm −x +P ,....,rm −x +P )
1 1 n n 1 1 1 n n n
= r(m +P ,....,m +P )
1 1 n n
.
Corollary 2.7. Let M be projective in σ[M] and semiprime. Suppose that
M satisfies ACC on left annihilators and P ,P ,...,P are the minimal prime
1 2 n
in M submodules, then
1. M ∼= E[M](M/P )⊕E[M](M/P )⊕...⊕E[M](M/P ).
1 2 n
2. Ifin addition M isa Goldie module, M/P has finite uniformdimension
c i
for all 0 ≤ i ≤ n.
n
Proof. 1. By Proposition 2.6 Ψ is a monomorphism and ImΨ ⊆ M/P .
e i
i=1
Thus we have the result. L
2. Since M is Goldie module, then M has finite uniform dimension By
(1) we have that M/P has finite uniform dimension for all 0 ≤ i ≤ n.
i
7
ThenextcorollarysaysthatthehypothesiseveryquotientM/P hasfinite
i
uniform dimension of [3, Proposition 2.25] can be removed.
Corollary 2.8. Let M be projective in σ[M] and semiprime. Suppose that
M has a finitely many minimal prime submodules P ,P ,...,P . Then M is
1 2 n
a Goldie module if and only if each M/P is a Goldie module.
i
Proof. Note that by Corollary 2.7.2 if M is a Goldie module then each quo-
tient M/P has finite uniform dimension, so it follows by [3, Proposition
i
2.25]
Lemma 2.9. Let M be projective in σ[M]. If M is a semiprime Goldie
module then Mn for all n > 0.
Proof. We have that σ[M] = σ[Mn] then Mn is projective in σ[Mn]. Since
M has finite uniform dimension then Mn so does. By [3, Theorem 2.8]
M is essentially compressible then Mn is essentially compressible by [15,
Proposition1.2]. Thusby[3,Lemma2.7andTheorem2.8]Mn isasemiprime
Goldie module.
Corollary 2.10. Let R be a ring. If R is a left Goldie ring then M (R) is
n
a right Goldie ring for all n > 0.
Proof. By Proposition 2.9 Rn is a semiprime Goldie module for all n > 0. By
[3, Theorem 2.22] End (Rn) ∼= M (R) is a semiprime right Goldie ring.
R n
Proposition 2.11. Let M be projective in σ[M]. Suppose M has nonzero
soclo. If M is a prime module and satisfies ACC on left annihilators then M
is semisimple artinian and FI-simple.
Proof. Let 0 6= m ∈ M. Since Zoc(M) 6= 0 and M is a prime module then
0 6= Zoc(M) Rm ⊆ Zoc(M) ∩ Rm. Hence Zoc(M) ≤ M. Let U be a
M e 1
minimal submodule of M. By [3, Corollary 1.17] U is a direct summand, so
1
M = U ⊕ V . If 0 6= V , since Zoc(M) ≤ M then there exists a minimal
1 1 1 e
submodule U ≤ V . Then M = U ⊕ U ⊕ V . If V 6= 0 there exists a
2 1 1 2 2 2
minimal submodule U ≤ V such that M = U ⊕U ⊕U ⊕V . Following in
3 3 1 2 3 3
this way, we get an ascending chain
U ≤ U ⊕U ≤ U ⊕U ⊕U ≤ ...
1 1 2 1 2 3
Notice that U ⊕...⊕U = Kerf where f is the endomorphism of M given
1 i
by
∼
M → M/(U ⊕...⊕U ) = V ֒→ M
1 i i
8
Thus, last chain is an ascending chain of annihilators, so it must stop in a
finite step. Then M is semisimple artinian.
Suppose that M = U ⊕...⊕U . If Hom (U ,U ) = 0 then U U = 0,
1 n R i j iM j
∼
but M is prime. Thus Hom (U ,U ) 6= 0, so U = U 1 ≤ i,j ≤ n.
R i j i j
Corollary 2.12. Let M be projective in σ[M]. Suppose M has essential
soclo. If M is a semiprime Goldie module then M is semisple artinian.
Proof. By Proposition 2.6 there exists an essential monomorphism Ψ : M →
M/P ⊕ ··· ⊕ M/P where P are the minimal prime in M submodules.
1 n i
Hence each M/P has nonzero socle and by Corollary 2.8 M/P is a Goldie
i i
module. Thus M/P is semisimple artinian and FI-simple for all 1 ≤ i ≤ n
i
∼
by Proposition 2.11. Then M = M/P ⊕···⊕M/P and hence semisimple
1 n
artinian.
Example 2.13. In Z−Mod, a module M projective in σ[M] is a semiprime
Goldie module if and only if M is semisimple artinian or M is free of finite
rank.
Proof. It is clear that a semisimple artinian module M is projective in σ[M]
and a semiprime Goldie module and by Proposition 2.9 every free module
of finite rank is a semiprime Goldie module. Now, suppose that M is a
Z
semiprime Goldie module and projective in σ[M]. Recall that in Z-Mod an
indecomposable injective module is isomorphic to Q or Zp∞ for some prime
p.
LetU beanuniformsubmoduleofM. BydefinitionE[M](U) = trM(E[Z](U)).
Suppose E[Z](U) ∼= Q, then E[M](U) ≤ Q. Since Q is FI-simple then
E[M](U) = Q. This implies that Q ֒→ M ∈ σ[M]. Hence σ[M] = Z−Mod.
Since M is projective in σ[M] = Z−Mod then M is a free module and since
M has finite uniform dimension then itchas finite rank.
Let n U ≤ M with U uniform. If one U is a torsion free group then
i=1 i e i i
M is free because above. So, we can suppose that every U is a torsion group.
i
L
Then, M hasessential soclo. ByCorollary2.12M is semisimple artinian.
Proposition 2.14. Let M be projective in σ[M] and semiprime. Suppose
that M is a Goldie Module and P ,P ,...,P are the minimal prime in M
1 2 n
submodules, then E[M](N ) ∼= E[M](M/P ) where N = Ann (P ). Moreover,
i i i M i
M/P contains an essential submodule isomorphic to N .
i i
9
Proof. By [6, Proposition 4.5], Ass (M/P ) = {P }. By Corollary 2.7.(1)
M i i
M ∼= E[M](M/P )⊕E[M](M/P )⊕...⊕E[M](M/P ). Then, following the
1 2 n
proof of [2, Theorem 2.20] there exist indecomposable injective modules
Fc,..,F in σ[M] and l ,...,l ∈ N such that M ∼= Fl1 ⊕...⊕Fln and each
1 n 1 n 1 n
E[M](M/P ) ∼= Fli. By[2,Theorem2.20]M ∼= Ek1 ⊕...⊕Ekn withE[M](N ) ∼=
i i 1 n i
Eki and k ,...,k ∈ N. Then by Krull-Remakc-Schmidt-Azumaya Theorem,
i 1 n
E ∼= F and l = k . Thus E[M](M/P ) ∼=cE[M](N ).
i i i i i i
Now, by Lemma 2.4, N ⊕P ⊆ M. Then there is an essential monomor-
i i e
phism N ֒→ M/P . In fact, N and P are pseudocomplements one of each
i i i i
other.
Lemma 2.15. Let M be projective in σ[M] and semiprime. Suppose that M
is a Goldie Module and P ,P ,...,P are the minimal prime in M submodules.
1 2 n
If N = Ann (P ), then Ann E[M](N ) = P for all 1 ≤ i ≤ n.
i M i M i i
Proof. Since N has finite unifor(cid:0)m dimens(cid:1)ion
i
E[M](N ) = E[M](U )⊕...⊕E[M](U )
i i1 iki
where U is an uniform submodule of N . By [2, Lemma 2.16], Ann (U ) =
ij i M ij
P for all 1 ≤ j ≤ k . Now by [2, Proposition 1.11 and Lemma 1.13],
i i
Ann (U ) = Ann (E[M](U )) for all 1 ≤ j ≤ k . Finally, by [6, Proposi-
M ij M ij i
tion 1.3], Ann (E[M](U )⊕...⊕E[M](U )) = ki Ann (E[M](U )) = P .
M i1 iki j=1 M ij i
Thus Ann E[M](N ) = P for all 1 ≤ i ≤ n.
M i i T
Proposition(cid:0) 2.16. Le(cid:1)t M be projective in σ[M] and semiprime. Suppose
that M is a Goldie Module and P ,P ,...,P are the minimal prime in M
1 2 n
submodules, then
1. Hom (E[M](M/P ),E[M](M/P )) = 0 if i 6= j.
R i j
2. If M is also a generator in σ[M] then E[M](M/P ) = M\/P .
i i
Proof. 1. It follows by [2, Proposition 2.21] and Proposition 2.14.
2. Since σ[M/P ] ⊆ σ[M] for every 1 ≤ i ≤ n, then M\/P ≤ E[M](M/P ).
i i i
It is enough to show that E[M](M/P ) ∈ σ[M/P ].
i i
We claim that P E[M](M/P ) = 0. If N = Ann (P ), by Lemma 2.15
iM i i M i
Ann (E[M](N )) = P . Now, by Proposition 2.14 E[M](N ) = E[M](M/P ).
M i i i i
Thus P E[M](M/P ) = 0.
iM i
By [5, Proposition 1.5] E[M](M/P ) ∈ σ[M/P ].
i i
10